(C2xC4).D12 - GroupNames

G = (C₂×C₄).D₁₂ order 192 = 2⁶·3

3^rd non-split extension by C₂×C₄ of D₁₂ acting via D₁₂/C₃=D₄

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C₂×C₄).₃D₁₂, (C₂×D₁₂).₃C₄, (C₂×C₁₂).₁₅D₄, (C₂×Q₈).₂₅D₆, C₄.₁₀D₄⋊₅S₃, (C₄×Dic₃).₁C₄, (C₆×Q₈).₁C₂², C₁₂.₁₀D₄⋊₁C₂, C₆.₁₃(C₂³⋊C₄), C₃⋊₁(C₄².C₄), C₂².₁₄(D₆⋊C₄), C₁₂.₂₃D₄.₁C₂, C₂.₁₄(C₂³.₆D₆), (C₂×C₄).₃(C₄×S₃), (C₂×C₁₂).₃(C₂×C₄), (C₂×C₄).₃(C₃⋊D₄), (C₂×C₆).₇(C₂²⋊C₄), (C₃×C₄.₁₀D₄)⋊₁₁C₂, SmallGroup(192,36)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₂ — (C₂×C₄).D₁₂

Chief series

C₁ — C₃ — C₆ — C₂×C₆ — C₂×C₁₂ — C₆×Q₈ — C₁₂.₂₃D₄ — (C₂×C₄).D₁₂

Lower central

C₃ — C₆ — C₂×C₆ — C₂×C₁₂ — (C₂×C₄).D₁₂

Upper central

C₁ — C₂ — C₂² — C₂×Q₈ — C₄.₁₀D₄

Generators and relations for (C₂×C₄).D₁₂
G = < a,b,c,d | a²=b⁴=1, c¹²=b², d²=b, cbc^-1=ab=ba, cac^-1=dad^-1=ab², bd=db, dcd^-1=b^-1c¹¹ >

Subgroups: 240 in 64 conjugacy classes, 21 normal (all characteristic)
C₁, C₂, C₂, C₃, C₄, C₂², C₂², S₃, C₆, C₆, C₈, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, Dic₃, C₁₂, D₆, C₂×C₆, C₄², C₂²⋊C₄, M₄(2), C₂×D₄, C₂×Q₈, C₃⋊C₈, C₂₄, D₁₂, C₂×Dic₃, C₂×C₁₂, C₃×Q₈, C₂²×S₃, C₄.₁₀D₄, C₄.₁₀D₄, C₄.₄D₄, C₄.Dic₃, C₄×Dic₃, D₆⋊C₄, C₃×M₄(2), C₂×D₁₂, C₆×Q₈, C₄².C₄, C₁₂.₁₀D₄, C₃×C₄.₁₀D₄, C₁₂.₂₃D₄, (C₂×C₄).D₁₂
Quotients: C₁, C₂, C₄, C₂², S₃, C₂×C₄, D₄, D₆, C₂²⋊C₄, C₄×S₃, D₁₂, C₃⋊D₄, C₂³⋊C₄, D₆⋊C₄, C₄².C₄, C₂³.₆D₆, (C₂×C₄).D₁₂

Character table of (C₂×C₄).D₁₂

class	1	2A	2B	2C	3	4A	4B	4C	4D	4E	6A	6B	8A	8B	8C	8D	12A	12B	12C	12D	24A	24B	24C	24D
size	1	1	2	24	2	4	4	4	12	12	2	4	8	8	24	24	4	4	8	8	8	8	8	8
ρ₁	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	1	1	1	1	1	1	1	1	1	-1	-1	-1	-1	1	1	1	1	-1	-1	-1	-1	linear of order 2
ρ₃	1	1	1	-1	1	1	1	1	-1	-1	1	1	1	1	-1	-1	1	1	1	1	1	1	1	1	linear of order 2
ρ₄	1	1	1	-1	1	1	1	1	-1	-1	1	1	-1	-1	1	1	1	1	1	1	-1	-1	-1	-1	linear of order 2
ρ₅	1	1	1	1	1	-1	-1	1	-1	-1	1	1	i	-i	-i	i	-1	-1	-1	1	-i	-i	i	i	linear of order 4
ρ₆	1	1	1	-1	1	-1	-1	1	1	1	1	1	i	-i	i	-i	-1	-1	-1	1	-i	-i	i	i	linear of order 4
ρ₇	1	1	1	1	1	-1	-1	1	-1	-1	1	1	-i	i	i	-i	-1	-1	-1	1	i	i	-i	-i	linear of order 4
ρ₈	1	1	1	-1	1	-1	-1	1	1	1	1	1	-i	i	-i	i	-1	-1	-1	1	i	i	-i	-i	linear of order 4
ρ₉	2	2	2	0	2	-2	2	-2	0	0	2	2	0	0	0	0	-2	-2	2	-2	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	2	0	-1	2	2	2	0	0	-1	-1	2	2	0	0	-1	-1	-1	-1	-1	-1	-1	-1	orthogonal lifted from S₃
ρ₁₁	2	2	2	0	-1	2	2	2	0	0	-1	-1	-2	-2	0	0	-1	-1	-1	-1	1	1	1	1	orthogonal lifted from D₆
ρ₁₂	2	2	2	0	2	2	-2	-2	0	0	2	2	0	0	0	0	2	2	-2	-2	0	0	0	0	orthogonal lifted from D₄
ρ₁₃	2	2	2	0	-1	-2	2	-2	0	0	-1	-1	0	0	0	0	1	1	-1	1	√3	-√3	√3	-√3	orthogonal lifted from D₁₂
ρ₁₄	2	2	2	0	-1	-2	2	-2	0	0	-1	-1	0	0	0	0	1	1	-1	1	-√3	√3	-√3	√3	orthogonal lifted from D₁₂
ρ₁₅	2	2	2	0	-1	-2	-2	2	0	0	-1	-1	-2i	2i	0	0	1	1	1	-1	-i	-i	i	i	complex lifted from C₄×S₃
ρ₁₆	2	2	2	0	-1	-2	-2	2	0	0	-1	-1	2i	-2i	0	0	1	1	1	-1	i	i	-i	-i	complex lifted from C₄×S₃
ρ₁₇	2	2	2	0	-1	2	-2	-2	0	0	-1	-1	0	0	0	0	-1	-1	1	1	-√-3	√-3	√-3	-√-3	complex lifted from C₃⋊D₄
ρ₁₈	2	2	2	0	-1	2	-2	-2	0	0	-1	-1	0	0	0	0	-1	-1	1	1	√-3	-√-3	-√-3	√-3	complex lifted from C₃⋊D₄
ρ₁₉	4	4	-4	0	4	0	0	0	0	0	4	-4	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₂³⋊C₄
ρ₂₀	4	4	-4	0	-2	0	0	0	0	0	-2	2	0	0	0	0	2√-3	-2√-3	0	0	0	0	0	0	complex lifted from C₂³.₆D₆
ρ₂₁	4	4	-4	0	-2	0	0	0	0	0	-2	2	0	0	0	0	-2√-3	2√-3	0	0	0	0	0	0	complex lifted from C₂³.₆D₆
ρ₂₂	4	-4	0	0	4	0	0	0	-2i	2i	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	complex lifted from C₄².C₄
ρ₂₃	4	-4	0	0	4	0	0	0	2i	-2i	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	complex lifted from C₄².C₄
ρ₂₄	8	-8	0	0	-4	0	0	0	0	0	4	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal faithful, Schur index 2

Smallest permutation representation of (C₂×C₄).D₁₂
►On 48 points

Generators in S₄₈

(2 14)(4 16)(6 18)(8 20)(10 22)(12 24)(26 38)(28 40)(30 42)(32 44)(34 46)(36 48)

(1 45 13 33)(2 46 14 34)(3 35 15 47)(4 36 16 48)(5 25 17 37)(6 26 18 38)(7 39 19 27)(8 40 20 28)(9 29 21 41)(10 30 22 42)(11 43 23 31)(12 44 24 32)

(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48)

(1 32 45 12 13 44 33 24)(2 11 46 43 14 23 34 31)(3 30 35 22 15 42 47 10)(4 21 36 41 16 9 48 29)(5 28 25 8 17 40 37 20)(6 7 26 39 18 19 38 27)

G:=sub<Sym(48)| (2,14)(4,16)(6,18)(8,20)(10,22)(12,24)(26,38)(28,40)(30,42)(32,44)(34,46)(36,48), (1,45,13,33)(2,46,14,34)(3,35,15,47)(4,36,16,48)(5,25,17,37)(6,26,18,38)(7,39,19,27)(8,40,20,28)(9,29,21,41)(10,30,22,42)(11,43,23,31)(12,44,24,32), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48), (1,32,45,12,13,44,33,24)(2,11,46,43,14,23,34,31)(3,30,35,22,15,42,47,10)(4,21,36,41,16,9,48,29)(5,28,25,8,17,40,37,20)(6,7,26,39,18,19,38,27)>;

G:=Group( (2,14)(4,16)(6,18)(8,20)(10,22)(12,24)(26,38)(28,40)(30,42)(32,44)(34,46)(36,48), (1,45,13,33)(2,46,14,34)(3,35,15,47)(4,36,16,48)(5,25,17,37)(6,26,18,38)(7,39,19,27)(8,40,20,28)(9,29,21,41)(10,30,22,42)(11,43,23,31)(12,44,24,32), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48), (1,32,45,12,13,44,33,24)(2,11,46,43,14,23,34,31)(3,30,35,22,15,42,47,10)(4,21,36,41,16,9,48,29)(5,28,25,8,17,40,37,20)(6,7,26,39,18,19,38,27) );

G=PermutationGroup([[(2,14),(4,16),(6,18),(8,20),(10,22),(12,24),(26,38),(28,40),(30,42),(32,44),(34,46),(36,48)], [(1,45,13,33),(2,46,14,34),(3,35,15,47),(4,36,16,48),(5,25,17,37),(6,26,18,38),(7,39,19,27),(8,40,20,28),(9,29,21,41),(10,30,22,42),(11,43,23,31),(12,44,24,32)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)], [(1,32,45,12,13,44,33,24),(2,11,46,43,14,23,34,31),(3,30,35,22,15,42,47,10),(4,21,36,41,16,9,48,29),(5,28,25,8,17,40,37,20),(6,7,26,39,18,19,38,27)]])

Matrix representation of (C₂×C₄).D₁₂ ►in GL₆(𝔽₇₃)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	1	0	72	0
0	0	1	0	0	72

1	0	0	0	0	0
0	1	0	0	0	0
0	0	46	54	0	0
0	0	0	27	0	0
0	0	46	27	0	27
0	0	46	27	27	0

0	1	0	0	0	0
72	1	0	0	0	0
0	0	1	0	71	0
0	0	0	0	72	1
0	0	60	0	72	0
0	0	14	46	72	0

72	1	0	0	0	0
0	1	0	0	0	0
0	0	1	0	71	0
0	0	1	0	72	72
0	0	14	46	72	0
0	0	60	0	72	0

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,1,1,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,46,0,46,46,0,0,54,27,27,27,0,0,0,0,0,27,0,0,0,0,27,0],[0,72,0,0,0,0,1,1,0,0,0,0,0,0,1,0,60,14,0,0,0,0,0,46,0,0,71,72,72,72,0,0,0,1,0,0],[72,0,0,0,0,0,1,1,0,0,0,0,0,0,1,1,14,60,0,0,0,0,46,0,0,0,71,72,72,72,0,0,0,72,0,0] >;

(C₂×C₄).D₁₂ in GAP, Magma, Sage, TeX

(C_2\times C_4).D_{12}

% in TeX

G:=Group("(C2xC4).D12");

// GroupNames label

G:=SmallGroup(192,36);

// by ID

G=gap.SmallGroup(192,36);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,141,36,422,184,1123,794,297,136,851,6278]);

// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=1,c^12=b^2,d^2=b,c*b*c^-1=a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,b*d=d*b,d*c*d^-1=b^-1*c^11>;

// generators/relations

Export

Character table of (C₂×C₄).D₁₂ in TeX

G = (C2×C4).D12 order 192 = 26·3

3rd non-split extension by C2×C4 of D12 acting via D12/C3=D4

G = (C₂×C₄).D₁₂ order 192 = 2⁶·3

3^rd non-split extension by C₂×C₄ of D₁₂ acting via D₁₂/C₃=D₄